The lid-driven cavity

The lid-driven cavity

Abstract The lid-driven cavity is an important fluid mechanical system serving as a benchmark for testing numerical methods and for studying fundamental aspects of incompressible flows in confined volumes which are driven by the tangential motion of a bounding wall. A comprehensive review is provided of lid-driven cavity flows focusing on the evolution of the flow as the Reynolds number is increased. Understanding the flow physics requires considering pure two-dimensional flows, flows which are periodic in one space direction as well as the full three-dimensional flow. The topics treated range from the characteristic singularities resulting from the discontinuous boundary conditions over flow instabilities and their numerical treatment to the transition to chaos in a fully confined cubical cavity. In addition, the streamline topology of two-dimensional time-dependent and of steady three-dimensional flows are covered, as well as turbulent flow in a square and in a fully confined lid-driven cube. Finally, an overview on various extensions of the lid-driven cavity is given.

One Introduction

A rectangular or a cubic container are among the most elementary confined geometries within which fluid motion can be studied. The simplest mechanical driving force acting on a viscous fluid with constant density and leaving the simple domain intact is the tangential in-plane motion of a bounding wall. A cuboid of which one of the solid walls moves tangentially to itself is called a lid-driven cavity.

Owing to the simplicity of its setup the lid-driven cavity has been investigated quite extensively. It has been employed as a numerical benchmark problem and as a test bed for studying particular physical effects. Searching the Web of Science for the topic lid-driven yields more than one thousand eight hundred hits. For these reasons, and because of the rapid evolution of this field of research, a review on lid-driven cavity flows seems justified, given that nearly twenty years have passed since the overview provided by Shankar and Deshpande.

After the first numerical investigations of Kawaguti and Burggraf the quest for efficiency and accuracy began with the work of Ghia et al. and Schreiber and Keller who computed the steady two-dimensional flow for Reynolds number up to one thousand four in a square cavity bounded by three rigid walls and a lid moving with constant velocity. Koseff and Street carried out a series of experiments on the flow in three-dimensional cavities with different lengths in the third dimension, many of them being summarized in Koseff and Street. Stimulated by these experimental results and the remaining open questions, dedicated three-dimensional test cases have been defined and investigated numerically by different research groups with results collected in Deville et al. After this joint effort, which did not yield very conclusive results for the targeted Reynolds number of Re equals three thousand two hundred, a new level of accuracy has been reached for two-dimensional flows by Botella and Peyret who employed spectral methods combined with a dedicated treatment of the singular corners/edges where the moving wall meets with a stationary wall. Their method yields highly accurate numerical solution for the two-dimensional problem up to Re equals one thousand three (see also Auteri et al.). With the progress in computing power and the routine computation of three-dimensional flows, benchmarks for three-dimensional flows became of interest. Applying the method of Botella and Peyret to three-dimensions Albensoeder and Kuhlmann provided highly accurate three-dimensional flow fields for Re equals one thousand three for different cavity lengths in the spanwise direction and for rigid and periodic boundary conditions at the end walls.

Apart from serving as a numerical benchmark, many fundamental fluid mechanical phenomena arise in the lid-driven cavity problem. An important aspect for an analytical and numerical treatment of the problem are the discontinuous boundary conditions along the edges at which moving and stationary walls meet. This problem is a special case of Taylor's scraping problem for which he has provided similarity solutions. Along such an edge with discontinuous boundary conditions for the velocity perpendicular to the edge, the vorticity and the pressure diverge at the apex. For two-dimensional flow, closed-form solutions have been obtained in terms of a series expansion of the steady